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$\large \begin{aligned} 51 \times 51 &= {\color{#3D99F6}26}{\color{#D61F06}01} \\ 52 \times 52 &= {\color{#3D99F6}27}{\color{#D61F06}04} \\ 53 \times 53 &= {\color{#3D99F6}28}{\color{#D61F06}09} \\ 54 \times 54 &= {\color{#3D99F6}29}{\color{#D61F06}16} \\ 55 \times 55 &= {\color{#3D99F6}30}{\color{#D61F06}25} \\ 56 \times 56 &= {\color{#3D99F6}31}{\color{#D61F06}36} \\ & \vdots \end{aligned}$
What is the value of $57 \times 57?$
Bonus: Can you explain why this pattern appears, and when it breaks?
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In the diagram, the yellow and red circles are identical and are both centered on the larger circle's diameter.
Which has a larger area?
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In a 90-question, multiple-choice test, a student gets
If the student attempted every single question, can he or she get a total score of exactly 300?
For example, if the student only got 60 correct answers (30 incorrect), then the total score would be $60\times4 - 30\times1= 210.$
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What mass (in kilograms) on the left side would keep the system in balance?
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We have two copper spheres of the same radius; one is hollow (vacuum inside) and the other is solid. Initially, both spheres are in a vacuum at the same temperature. Then they are heated up to the same temperature which causes them both to expand.
Which sphere has the greater final radius?
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